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On the spectrum of the Steklov problem in a domain with a peak. (English. Russian original) Zbl 1171.35086
Vestn. St. Petersbg. Univ., Math. 41, No. 1, 45-52 (2008); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 2008, No. 1, 56-65 (2008).
Consider the spectral Steklov problem \[ -\Delta_xu(x)=0\quad (x\in \Omega ),\quad \partial_N u(x)=\lambda u(x)\quad (x\in\partial \Omega\setminus \{\mathcal O\}), \tag{1} \] where \(\Omega\) is a domain in \(\mathbb R^n\), \(n\geq 2\) , with a compact boundary \(\partial \Omega\); suppose \(\partial \Omega\) is smooth everywhere except the origin \(\mathcal O\) of the Cartesian coordinate system \(x=(y,z)\in\mathbb R^{n-1}\times\mathbb R\). In a neighborhood of point \(\mathcal O\), domain \(\Omega\) has a peak and is defined by the relations \(z>0\), \(z^{-1-\gamma}y\in \omega\), where \(\gamma >0\) is the cusp factor and \(\omega\) is a domain in \(\mathbb R^{n-1}\) with a smooth boundary \(\partial\omega\); \(\Delta_x\) is the Laplace operator and \(\partial_N\) is the derivative along the outer normal.
In the paper the problem (1) is examined in terms of the Hilbert space \(\mathcal H\); this space is introduced as a subspace of the Sobolev space \(H^1(\Omega)\) equipped with the norm \(\| u;\mathcal H\|=(\|\nabla_xu;L_2(\Omega)\|^2+\|u;L_2(\partial\Omega)\|^2)^{1/2}\).
Put \(\lambda_\#=(n-3/2)^2\)mes\(_{n-1}(\omega)/\)mes\(_{n-2}(\partial\omega)\). The main result of this paper is the following.
Theorem. The spectrum of problem (1) is located on \([0,+\infty)\). 1) \(\forall \gamma\in(0,1)\) the spectrum is discrete and the eigenvalues form an infinitely increasing sequence \(0=\lambda_1\leq\lambda_2\leq\dots\leq\lambda_k\leq\dots\longrightarrow +\infty\). 2) If \(\gamma=1\), then the spectrum is discrete on \([0,\lambda_\#)\); the ray \([\lambda_\#,+\infty)\) constitutes the continuous spectrum. 3) If \(\gamma >1\), then point \(\lambda =0\) belongs to the continuous spectrum of problem (1).

35P05 General topics in linear spectral theory for PDEs
Full Text: DOI
[1] O. A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics (Nauka, Moscow, 1973; Springer, Berlin, 1985). · Zbl 0169.00206
[2] V. G. Maz’ya, Sobolev Spaces (Leningrad. Gos. Univ., Leningrad, 1984; Springer, Berlin, 1985).
[3] E. Kamke, Differentialgleichungen Lösungsmethoden und Lösungen (B.G. Teubner, Stuttgart, 1983), Vol. 1, 10 Aufl.
[4] M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators in Hilbert Space (Leningrad Gos. Univ., Leningrad, 1980; Reidel, Dordrecht, 1987).
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